Cauchy difference operator in \({\mathcal L}^p\)
Let \(E\) be a Banach space, \((G,\cdot,\Sigma,\lambda)\) a complete measurable group with \(\lambda\) (left or right)-invariant, and for \(f:G\to E\) define the Cauchy difference \({\mathcal C}f:G\times G\to E\) by \(({\mathcal C}f)(x,y) := f(xy) - f(x)-f(y)\). Under which conditions on \({\mathcal C}f\) is \(f\) close to an additive function? Under the assumption that \(\lambda(G)=\infty\), \textit{Józef Tabor} [Result. Math. 32, 145-158 (1997)] proved that if \({\mathcal C}f \in {\mathcal L}^p_{\lambda\times\lambda}(G\times G,E)\), then there is a \(K>0\) and an additive \(A:G\to E\) with \(| | f-A| | _p \leq K | | {\mathcal C} f| | _p\). The present paper treats the case \(0<\lambda(G)<\infty\). One of the main results is the same conclusion as above, with \(K=\lambda(G)^{-1/p}\). Another result concerns the case where \(\cdot\) is an arbitrary binary operation on \(G\): If \(f \in L^p_\lambda(G,E)\) and \({\mathcal C}f\) is measurable, then this conclusion again holds, in fact with \(A\equiv 0\). In this case the linear operator \(\mathcal C\) is continuous on an appropriate domain and continuously invertible on its image. This extends some results by \textit{E. V. Shulman} [J. Lond. Math. Soc., II. Ser. 54, 111-120 (1996; Zbl 0861.39020)], but only in the special case of the Cauchy equation on compact groups.
scientific article; zbMATH DE number 1123839